A new numerical method is presented for solving problems of one-dimensional fluid flows in pipelines with significant thermal effects. The method can be considered as a generalization to straightforward collocation. Pressure and velocity are still approximated by continuous piecewise-linear functions, but temperature is approximated with velocity-upwinded picewise-discontinuous constants. This certainly contributes to the stability problems, but substantially increases the complexity of the analysis. Theorems on asymptotic convergence of the method have been established, computational illustrations have been given, and possible extensions of the study have been pointed out.